Confidence with the use of blog

Virtual Classroom by Fariel Mohan
Did the virtual classroom helped improved your confidence in Maths?
Has the virtual classroom help your understanding in Maths?
Would you recommend a student to use this virtual classroom?
Do you feel you were included and involved when you answered or asked questions?
Was the flexibility of using at any time significant to you?
In explaining or making a comment, did you ever had to stop and ask yourself how can I really explain what I am trying to say?
Did you ever felt good at helping someone by correcting their error or suggesting what might be the error?
Did you feel you were included and involved in the teaching of this course?
Being able to ask a question whenever you want, was this useful?
Was it helpful to you to use anonymous names?

Trig Calculus

1. ∫5sinb db
2. ∫5cosm dm
3. ∫5sinb + 9b^4 db
4. ∫5/x dx
5. Find the derivative of the following

Let f(t) = 5 cos t , f(x) = sin x, Let f(x) = 3cos x + 5x, Let f(t) = 2t + sin t. f(t) = 1/t2 - sin t, Let f(x) = sin x - cos x. Let f(x) = 2sin x + 3cos x. Let f(x) = 4sin x + 1/x.
Let f(t) = -3sin t + 1/2 cos t.

6. Find the integration of the following
Let f(t) = 5 cos t , f(x) = sin x, Let f(x) = 3cos x + 5x, Let f(t) = 2t + sin t. f(t) = 1/t2 - sin t, Let f(x) = sin x - cos x. Let f(x) = 2sin x + 3cos x. Let f(x) = 4sin x + 1/x. Let f(t) = -3sin t + 1/2 cos t.

Real Life ie rate of change

1. The displacement s of a piston during each 8-s is given by s = 8t -t^2. For what value of t is the velocity of the piston 4?
2. The distance s travelled by a subway train after the brakes are applied is given by s = 20t -2t^2 How far does it travel after the brakes are applied in coming to a stop?
3. Water is being drained from a pond such that the volume V of water in the pond after t hours is given by V = 50t(60-t^2). Find the rate at which the pond is being drained after 4 hours.
4. The electric field E at a distance r from a point charge is E=k/r^2 where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.
5. The altitude h of a certain rocket as a function of the time t after launching is given by h = 550t - 4.9 t^2. What is the maximum altitude the rocket attains?
6. The blade of a saber saw moves vertically up and down and its displacement is given by y = 1.85 sin t. Find the velocity of the blade for t=0.2
7. Differentiate y = 7sin x + 4x^2 +1

Review Questions

Integrate the following DO NOT FORGET THE +c
1. dy/dx = 5x^3 + 2x^2 + 5
2. dy/dx = 6x + 1
3. dy/dx = 2
4. dy/dx = 8x^5 - 5x^3 + 4x
5. dy/dx = 7x^2 + 3x + 4

6. A particle moves in a straight line and at point P, it's velocity is given as v = 7t^2 - 5t +3. The particle comes to rest at point Q.1. What is the acceleration at Q if it arrives at Q when t=7?2. How far does the particle travel in t=1 to t=4?

7. Find dy/dx of the following:

1. y = 6x^3 + 4x^2 - 5x
2. y = x^-3 + 16
3. y = 4x + 4
4. y = 3x^4 + 3(x^2 + 5)
5. y + 6 = 4x + 9y - 3x^2


8. A curve has equation y = 4/(2)^.5, find dy/dx
9. A curve is such that dy/dx = 16/x^3 and (1,4) is a point on the curve, find the equation of the curve.
10. y = 6theta - 2sin theta, find dy/dx
11. The equation of a curve is y = 2x + 8/x^2, find dy/dx and d^2ydx^2
12. A curve is such that dy/dx = 2x^2 -5. Given that the point (3,8) lies on the curve, find the equation of the curve.

Questions

1. A rectangle is to have a perimeter 26cm and a length x cm. If the width x equals 3cm, find the length and hence the Area of the rectangle. Find dA/dx and hence find the width of the rectangle giving the maximum area.
2. Solve for p and r given 3p + 2r = 7 and p^2 - 2r = 11
3. Solve 8x^2 + 3y^2 = 50 and 2x + y = 5
4. Write the expression 9x^2 - 9x + 1 in the form a(x + b)^2 + c, where a,b and c are constants. Hence state whether the function y = 9x^2 - 9x + 1 has a maximun or minimum value. State the value of x at which this maximum or minimum value occurs
5. EFGH is a parallelogram with EF = 6cm EH = 4.2cm, and angle FEH = 70 degrees. Calculate the length of HF. Calculate the area of the parallelogram EFGH.
6. A vertical tover FT has a vertical antenna TW mounted on top of the tower. A point P is on the same horizontal ground as F such that PF = 28 m and the angle of elevation of T and W from P are 40 and 54 resp. Calculate the length of the antenna TW.
7. A vertical pole AD and a vertical tower BC stands on horizontal ground XABY. The height of the pole is 2.5 m and the angle of depressionn of B from D is 15 degrees abd the angle of elevation of C from D is 20 degrees. DE is a horizontal line. Calculate AB and the height of the tower BC.
8. Find the coordinates of the stationary points on the graph of y = x^3 - 12 x - 12.
9. Factorise x^2 + x -12
10. Evaluate (5x^2 + 4x + 1) + (-7x + 2)
11. Factorise 5x^2 + 13x - 6
12. Simplify (8x^4 - 2x^2)/2x^2
13. Factorise 3x^2 - 48
14. KNM is a right angle triangle with KNL also being a right angle triangle. AN = 6 cm and NM = 15.6 cm and angle KLN is 52 degrees. KLM is a straight line. Calculate the size of angle KMN and the length LM
15. F(x) = 3x^2 - 12x + 5. Write in the form a(x + b)^2 + c, where a,b and c are constants. Hence determine the minimum value of f(x) and the coordinates of the minimum point.
16. y = 6x^2 + 32/x^3, find dy/dx
17. y = x^2 + 54/x, find dy/dx
18. y = 1 + 4x^3, find dy/dx
19. Find the coordinates of the stationary points on the graph of y = x^2 + x^3.
20. The curve y = 27 - x^2 has the points P and S on the curve. The point R and Q lie on the x-axis and PQRS is a rectangle. The length of OQ is t units. Find the length of PQ in terms of t and show the area of PQRS is A = 54t - 2t^3. Find the value of t for which A has a stationary value. What is the stationary value of A and determine its nature (max or min)

Graphs Project

The graph project was an individual project, if group the quantities must be multiplied, so 4457, 255, 133, 2282, 4420 redo or the project which is 1 student will be divided by 5.

This project was suppose to be the 3 sections:

• Show a graph being used in real life for 3 cases
• In graphs line and curve identify keep points and explain what they signify
• Show how 2 graphs can be used to solve (intersection) and take some points from a lab or internet and plot the graph

Generally speaking very little time was spent on this project so it is reflected. If anyone wants to add to their project, do so before start of exam (mon)

Leave with admin assistant and write added to project

So far 60 projects was collected

So far 84 blog users and 1 manual user

So far 95 quiz were done

Trigs

Trig deals with 3 lengths placed together to form an enclosed shape called a triangle. This triangle will now have 3 angles. Remember an angle can be formed with 2 lengths like a Vand this angle will create a length which is like the open side of the V.
A triangle can be of 2 types:

• right angle
• non right angle

With the right angle triangle, one angle is 90 degrees. The sides of the triangles are labelled or named with respect to an angle (not the 90) in the triangle.

• Identify the angle
• Longest side hyp
• opp is opposite to the angle
• adj is adjacent to the angle

Anything to be found can be evaluated using sin, cos, tan or pythagoras theorem.

With a non right angle triangle

• label the vertices
• label the angles
• Label the length associated with the angle

Use sine rule or cosine rule

In trig questions it is important to identify all the triangles in the figure and draw the separately

1. DEFG is a trapezium with DX a perpedicular line to GF. DE = 10cm, DG = 13cm, Angle EFX and DXF are right angles. Find the length DX. The area of the trapezium.
2. ABC and PCD are right angle triangles. Angle ABC = 40 degrees, AB = 10 cm, PD = 8 cm and BD = 15cm. Angles BCA and DCP are right angles. Find the length BC the angle PDC.
3. A plane takes off at an angle of elevation 17 degrees to the ground. After 25 seconds the plane has travelled a horizontal distance of 1400. Calculate the height of the plane above the ground after 25 seconds.
4. STW is a triangle with the angle STW is 52 degrees. ST = 5cm, TW is 9cm. Calculate the lenght = SW and the area of triangle STW.

Calculus Revision questions

Differentiation deals with change change of something with respect to another thing
y = 4 has no change since 4 is a constant
y = 4x is a straight line which means steady or fixed change y = mx + c so fixed change is gradient which is m which is 4
y = 4x^2 means this is a curve so there is change at every point(different grad at every point) so calculas helps in approximating by giving the following rate of change

• y=ax^n dy/dx = anx^(n-1)
• y = sin x dy/dx = cos x
• y = cos x dy/dx = - sin x

If gradient or tangent or normal is to be found, it must be at a specific point since each point has its own gradient or tangent or normal.

Polynomial differentiation

1. Find dy/dx of the equation y = 2x + 8/(x^2). Find the coordinates of the turning point, is it max or min and verify your answer. Find the normal to the curve AT THE POINT (-2, -2)
2. A curve has equation y = k/x where k is a constant. Given that the gradient of the curve is -3 when x = -2, find the value of the constant k.
3. The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the curve and determine the nature of the turning points.
4. The equation of a curve is y = x^3 - 8. Find the equation of the normal to the curve AT THE POINT where the curve crosses the x-axis.
5. A curve is such that dy/dx = 16/x^3 and (1, 4) is a point on the curve. Find the equation of the curve.
6. A curve is such that dy/dx = 2x^2 - 5 and (3, 8) is a point on the curve. Find the equation of the curve.

Integration deals with area. Area must be bounded otherwise area cannot be found. Hence to be bounded, normally 4 lines must be known to give an enclosed region. Integration of a curve gives the area under the curved trapped towards the x-axis. Two other lines are given by the lines x = 7 to x = 1 which is called upper bound and lower bound. WHen 2 curves are given the bounds are found by determining the points of intersection.

1. The curve y = 3x^.5 and the line y = x, find the two x bounds (ie the points of intersection) Find the area under the curve but above the line.
2. The region P is bounded by the curve y = 5x - x^2, the x=axis and the line x = h. The region Q is bounded by the curve y = 5x - x^2, the xaxis and the lines x = h and x = 2h. Given that the area of Q is twice the area of P, find the value of h.

Revision questions

1. f(x) = 2 - 3x -2x^2 bring to the form a(x + b)^2 + c and hence determine the coordinates of the turning point. Is the t.p. a max or min justify your answer.
2. Solve the equations 3x +y = 14 and 2x^2 -xy = 3
3. Express 1 - 4x - 2x^2 in the form a - b(x + c) hence when ix the function a maximum
4. CDB is a right angle triangle, BC is 5 metres and BCD is 40 degrees and BDC is 90 degrees. Calculate the length of BD and DC. Show that the area of the triangle is 12.5sin 40cos40.
5. Solve the equation 5y^2 = 8y - 2
6. MLO is a triangle with MNL a right angle triangle and NLO another right angle triangle. ML is 26 cm, NL is 10 cm, MOL is 35 degrees. Calculate the length of MN and MO.
7. Solve the simultaneous equations 2x^2 + y^2 = 33 and x + y = 3
8. Solve the simultaneous equations x + 1 = 2y and x^2 - 3y = 4
9. EFGH is a parallelogram EFI is 40 degrees, EF = 8m, EI is a perpendicular line to FG such that IG = 5m. Calculate the length FI, EI and the area of EFGH.
10. ABCD is a trapezium. AB = 12m, AD = 1.5m, BC = 3m and AD is parallel to DC. I is a point on BC such that CID is 90 degrees. Calculate the angle CDI and the length of DC.

Maths quiz mark is out

Marks were not very good
Please check with admin assistant or class rep
If anyone would like to reattempt a quiz
to be instead of their current mark
This wednesday 2nd Dec I will give a chance
must be from 9:00 am - 12:00 noon
Revise and redo
It will be in my staff room

Also see me on blog names
out of the 79, I have the real name for 65

What about the rest of the class (120 -79), your deadline for submission
was fri 27 or mon 30 with your at least 60 comments

Do not forget graph projects due nov 30th

Integration

1. Integrate (1 + 2 sin x ) wrt x with 45 and 0 degrees as bounds8.
2. Integrate 2x + 3 cos x ) wrt x with 90 and 60 degrees as bounds

These questions has bounds of degrees since trig are involved.

1. Integrating term by term gives,
2. integrate 1 wrt x and integrate 2sin x wrt x
3. this gives x - 2cos x + c
4. Now the bounds are used to get the required bounded region
5. Because the curve is a trig, degrees or radians can be used
6. In this case the curve has a trig part and a polynomial part so radians must be used

1. ABCD of a swimming pool, 12 m long. AB is the horizontal top edge. AD is the depth of the shallow end and BC the depth of the deep end.
i) Calculate the angle that DC makes with the horizontal
ii) Calculate the length of the sloping edge, DC
iii) If the pool is 5m wide, calculate the total surface area.

2.One face of the roof of a house in the shape of a parallelogram ABCD. The angle ABF= 40 degrees and the length AB = 8m. AI represents the rafter placed perpendicular to BC such that FC = 5m.
Calculate i) the length BI
ii) the length AI
iii) the area of ABCD

3. Differentiate the following i) x + 1/x
ii) 2cos x – sin x + 3x^7+ x^4
iii) √(x^3) +11x + 7
b) Integrate 3x^5 – 2sin x dx

Last class questions

1. Solve the following system of equations.
   x^2 + 4y^2 = 20
   xy = 4
2. A vector v has a magnitude of 200m and an angle of elevation of 50o. To the nearest tenth of a meter, find the magnitudes of the horizontal and vertical components of v.
3. Find the area bounded by the region y = x^2 + 1 , y = -x + 3, x = 0 and x = 3
4. Find the derivative of y = x^2 + 1/x
5. Find the derivative of y = 1 + 4x^3
6. Find the derivative of y = x^2 + 54/x
7. Displacement is given by s = 2 + 3t - t^2, find the maximum displacement
8. Find the coordinates of the stationary point of y = x^3 -12x - 12
9. i) Factorise by trial and error the expression f(x) = 2x^2 + x + 15 ii) Write in the form f(x) = a(x + b)^2 + c iii) Hence find the turning point of f(x) iv) Differentiate 2x^2 + x + 15 hence determine the turning point. v)Comment on iii) and iv) v1) Sketch the curve 2x^2 + x + 15 and determine the factors of the curve and the turning point vii) Using te graph solve 2x^2 + x + 15 = 4
10. The curve y^2 = 12x intersects the line 3y = 4x + 6 at 2 points. Find the distance between the 2 points.

1. Two ropes hold a boat at a dock. The tension in the ropes can be represented by 40 + 10j N and 50 – 25j N. Find the resultant force.
2. The total power P (in watts) transmitted by an AM radio station is given by
   
   P = 500 + 250 m^2 ,
   where m is the modulation index. Find the instantaneous rate of change of P with respect to m for m = 0.92
3. Find the derivative of the following polynomials:
   i. y = 4 x^2 + 7 x + 3
   ii. y = 4 x^ -6 - 5 x^3 + x
4. Find the derivative of the following trigs:
   a. y = 6 sin Ө
   b. y = 7 sin Ө + 3 cos Ө
5. Determine the gradient of each of the given functions at the given point.
   a) s = 2t^3 - 5t^2 + 4 (-1,-3)
   b) y = 5 sin Ө where Ө = 38◦
6. Find the area under the curve y = x^3 that is between the lines x = 1 and x = 2.
   
7. Integrate the following
   i) 4x^6 + 3x^ -4 + 1
   ii) 2 cos Ө
8. The electric field E at a distance r from a point charge is E = k/(r2), where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.
9. The blade of a saber saw moves vertically up and down, and its displacement y (in cm) is given by y = 1.85 sin t, where t is the time (in s). Find the velocity of the blade for t = 0.025 s.
10. An open-top container is to be made from a rectangular piece of cardboard 24 cm by 38 cm. Equal squares of side x cm are to be cut from each corner, then the sides are to be bent up and taped together. Find the instantaneous rate of change of the volume V of the container with respect to x for x = 4 cm.
11. An analysis of a company’s records shows that in a day the rate of change of profit (in dollars) in producing x generators is dp/dx = 60(30 – 4x^5). Find the profit in producing x generators if a loss of $500 is incurred if none are produced.
12. The vertical displacement y (in cm) of the end of an industrial robot arm for each cycle is y = 2t^1.5 – cos t, where t is the time (in s). Find its vertical velocity for t = 15s.

Maths Problems set #3

17. Solve for x and y
x^2 = 4 – y x = y + 2

18. Expand the following
i) 4x(3x +2)
ii) (4x + 7)(4x + 7)
iii) (4x + 7)(3x +2)
b) Find the roots of the equation 2x^2 – 11x -21 = 0

19. Solve, for 0◦ < Ө < 360◦, giving your answer to 1 decimal place where appropriate
i) 2 sin Ө = 3 cos Ө
ii) 2 – cos Ө = 2 sin^2 Ө

20.Using lasers, a surveyor makes the measurements shown below, where B and C are in a marsh. The angle PAC is 90 degrees. Find the distance between B and C.
PA = 265.74
angle APB 21.66 degrees
angle APC 8.85 degrees

Questions 1

10. If Integrate (4x + k ) wrt x with 2 and 1 as bounds = 1 find k

11. Find the area of the region enclosed by y = 4 – x ^2 and the x –axis
from x = -1 to x = 1
12. The amount of liquid V cm ^3 in a leaking tank at time t secs. is given

by V = ( 20 – t ) ^3 for 0 ≤ t ≤ 20
Find the rate at which water leaves the tank when t = 5 secs.
13. Given 3y –x + 6 = 0
(1) Make y the subject of the formula 3x - y + 6 - 0
State the gradient, and the y – intercept
14. Using any suitable method , solve the quadratic equation

2x^2 – 4x + 1 = 0
( give answers to 3 s.f )
c) The floor of a room is in the shape of rectangle . The floor is ‘ C ‘ metres long. The width is 2 metres less tha its length .
(1) State in terms of C
(a) the width of the floor
(b) the area of the floor

(ii ) If the area of the floor is 15 m ^2 write an equation in C to show the information.
(iii) Use the equation to determine the width of the floor

15 Solve for p and r given

3p + 2r = 7
p2 – 2r = 11

1 Find dy/dx if y = 4x ^3 - 3 x^ 2 + 2x
2 Differentiate with respect to x

y = 5 x ^3 - 2 x^ -2 + 1

(3) Find dy/dx at x = ñ/8 if y = 6 x^ 2 + 2 sin x

(4) Differentiate , with respect to ø if

v = 2ø ^ 3 + 2 cos ø + sin ø

(5) Integratewith respect to x

2 x^ 4 + 4/x^ 2+ x

6. Integrate w.r.t. x
2 x ^2( 2x + 3 ) Hint Expand then integrate

7 Integrate (1 + 2 sin x ) wrt x with 45 and 0 degrees as bounds

8. Integrate 2x + 3 cos x ) wrt x with 90 and 60 degrees as bounds

Area using Integration

Explain how you would find the area under a curve by using integration.

What is f(x)? What is a and b?

Triangles

All triangles consists of 3 lengths and 3 angles.

What are the 2 types of triangles?
For each type state the properties or rules that can be applies.
Illustrate with examples

Completing the square Vs differentiation

Use the differentiation approach and the completing the square approach
and comment on your observations

Find the coordinate of the turning point and determine whether it is a maximum or a minimum
and what is its maximum or minimum value.

1. y = x^2 + x -6
2. y = x^2 + 14x -5
3. y = x^2 -23x + 18
4. y = 2x^2 - 8x + 5
5. y = 5x^2 -30x + 12
6. y = 8x^2 -32x + 25

Examining differentiation

Differentiation provides a way to get an approximation for gradient at a given point.

Consider the curve
y= x^2 +6x + 8

Differentiating to check gradient at points is:
dy/dx = 2x + 6
This is the gradient machine, can now find gradient for any point

What is the gradient at a turning point?
It is 0. Explain why?

At what point is this curve turning
when gradient = 0
2x + 6 = 0
so x = -3
curve turns when x = -3
What is the coordinate for this turning point?

when x = -3, what is y
substitute x =-3 into the curve y= x^2 +6x + 8
Show that the co-ordinate is (-3, 1)

Now take the same curve y= x^2 +6x + 8
and complete the square into the form a(x +b)^2 + c
show that the turning point is -3 and that the curve is a minimum with a value of 1.

Relating graphs with other topics in maths

What are common uses of graphs?
Can simultaneous equations be solved by graphs and why?
What does the x-intercept mean that is what is the meaning of y = 0?
Can factoring of quadratic give the x-intercepts?
What does completing the square useful for?
Explain how differentiation can be used to find turning point?
What really does differentiation gives?

Graphs

What really is a graph?
What is a graph used for?
What are the key things to look for in a graph?
What are some other ways of saying x-intercept?
Can you determine or have an idea of how many x-intercepts may be present?

What do these really mean

Explain what is being said in these equations:

1. m = 6 + 13
2. m = 5b + 2
3. h = irs
4. w = 5(7d - 2)
5. p = hv + k
6. y = mx + c
7. g = h/3 + t
8. p = (7f - 9)/5
9. h = (5m + 3) /p
10. m = (hp + 3)5

Searching for fractions 100 different ways

Identify any use of fractions from your day so far?
Why do you associate fraction with that scenario?
What would happen if a whole number was used in your scenario instead of a fraction?

Exploring Fractions

What is a fraction?
What really is the purpose of using fractions?
How can a fraction be identified?
How will you exppalin about fractions to your 9 years old brother?
Explain how necessary are fractions to us?

Revision question for quiz

1. a.) For a given spring, F has the value 35 when the spring has stretched 8 inches. What is the constant of proportionality for that spring?
b) What is the value of F when the spring has stretched 11 inches?

2. The distance d that an automobile travels varies directly as the time ta) that it travels. After 2 hours, the car has traveled 115 miles.
b) Write the equation that relates d and t.b) The units on the right must be same as those of d on the left, that is, a) distance. What are the units, then, of 57.5?c) How far has the car traveled after 7 hours?

3. In a microwave receiver circuit, the resistance R of a wire 1m long is given by R = k/d2, where d is the diameter of the wire. Find R if k = 0.000 000 021 96 Ω m2 and d = 0.000 079 98m.

4. A computer can do an addition in 7.5 x 10^-15 s. How long does it take to perform 5.6 x 10^6 additions?

5. Show by first principle that the differential of y = 4x^2 -3x is 8x.

6. Water leaked into a gasoline storage tank at an oil refinery. Finding the pressure in the tank leads to the expression -4(b – c) – 3(a – b). Simplify this expression.

7. A shipment contain x film cartridges for 15 exposures each and x + 10 cartridges for 25 exposures each. What is the total number of photographs that can be taken with the film from the shipment?

8. In determining the size of a V belt to be used with an engine, the expression 3D – (D – d) is used. Simplify this expression.

9. When finding the current in a transistor circuit, the expression i1 – (2 – 3 i2) + i2 is used. Simplify this expression.

10. Each of two stores has 2n + 1 mouse pads costing $3 each and n – 2 mouse pads costing $2 each. How much more is the total value of the $3 mouse pads than te $2 mouse pads in the two stores?

11. In designing a certain machine part, it is necessary to perform the following simplification
16(8 – x) – 2(8x – x^2) – (64 – 16x + x^2)
What will be result be?

12. When analyzing the potential energy associated with gravitational forces, the expression
(GMm[(R + r) – (R – r)]) /2rR
arises. Preform the indicated division.

13. In the optical theory dealing with lasers, the following expression arises:
(8A^5 + 4A^3µ^4E^4)/8A^4
Perform the indicated division.

14. In finding the total resistance of the resistors, the expression
(6R1 + 6R2 + R1R2)/ 6R1R2
is used. Perform the indicated division.

15. The ratio of electric current I (in A) to the voltage V across a resistor is constant. If I = 1.52A for V= 60.0V, find I for V = 82.0V.

16.In order to find the distance such that weights are balanced on a lever, the equation
210(3x) = 55.3x + 38.5(8.25 – 3x)
must be solved. Find x.

17. In the study of the forces on a certain beam, the equation
W = L(wL + 2P)/8
is used. Solve for P.

18. A medical researcher finds that a given sample of an experimental drug can be divided into 4 more slides with 5mg each than with 6mg each. How many slides with 5mg each can be made up?

Variation

When a varies directly as b, we often say, "a is proportional to b." In that case, the relationship between a and b takes this algebraic form:
a = kb.
k is called the constant of proportionality.
The circumference of a circle, for example, varies directly as the diameter. The constant of proportionality is called π.
C = πD.

Example
a.) For a given spring, F has the value 35 when the spring has stretched 8 inches. What is the constant of proportionality for that spring?
b) What is the value of F when the spring has stretched 11 inches?

a) The distance d that an automobile travels varies directly as the time ta) that it travels. After 2 hours, the car has traveled 115 miles. Writea) the equation that relates d and t.
b) The units on the right must be same as those of d on the left, that is, a) distance. What are the units, then, of 57.5?
c) How far has the car traveled after 7 hours?

3.) If the side of a square doubles, how will the perimeter change?

4) a varies as the square of b. When b = 7, a = 4. What is the value of a when b = 35?
a varies as the square of b. When b = 20, a = 32. What is the value of a when b = 15?

5) The area A of a circle varies directly as the area of the circumscribed square. That is, as the area of the square changes, the area
of the circle changes proportionally.
a) Show that this implies that the area A of the circle varies as the square of the radius r.
b) If the radius of a circle changes from 6 cm to 12 cm, how will the area change?
c) What is the constant of proportionality that relates the area A to r²?

Simultaneous Equations

Why is 2 variables connected?
Why is 2 equations with the same 2 variables required to solve for the variables?

1. 1000 tickets were sold. Adult tickets cost $8.50, children's cost $4.50, and a total of $7300 was collected. How many tickets of each kind were sold?
2. Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate?
3. Samantha has 30 coins, quarters and dimes, which total $5.70. How many of each does she have?
4. "36 gallons of a 25% alcohol solution"
   means: 25%, or one quarter, of the solution is pure alcohol.
   One quarter of 36 is 9. That solution contains 9 gallons of pure alcohol.
   Here is the problem:
   How many gallons of 30% alcohol solution and how many of 60% alcohol solution must be mixed to produce 18 gallons of 50% solution?
   "18 gallons of 50% solution" means: 50%, or half, is pure alcohol. The final solution, then, will have 9 gallons of pure alcohol.
5. A saline solution is 20% salt. How much water must you add to how much saline solution, in order to dilute it to 8 gallons of 15% solution?
6. It takes 3 hours for a boat to travel 27 miles upstream. The same boat can travel 30 miles downstream in 2 hours. Find the speeds of the boat and the current.
7. A total of 925 tickets were sold for $5,925. If adult tickets cost $7.50, and children's tickets cost $3.00, how many tickets of each kind were sold?
8. Mr. B. has $20,000 to invest. He invests part at 6%, the rest at 7%, and he earns $1,280 interest. How much did he invest at each rate?
9. How many gallons of 20% alcohol solution and how many of 50% alcohol solution must be mixed to produce 9 gallons of 30% alcohol solution?
10. 15 gallons of 16% disenfectant solution is to be made from 20% and 14% solutions. How much of those solutions should be used?
11. It takes a boat 2 hours to travel 24 miles downstream and 3 hours to travel 18 miles upstream. What is the speed of the boat in still water, and how fast is the current?
12. An airplane covers a distance of 1500 miles in 3 hours
    when it flies with the wind, and in 3 1/3 hours when it flies against the wind. What is the speed of the plane in still air?

Exponent teaser

1. Solve for x and y

2^(x - 1) * 3(y + 1) = 72 * 6^(y - x + 2)

3^(x - 1) * 2^(y + 1) = 3 * 6^(x - y - 2)

2. Find the error in this proof. Prove: 5 = 7

Let a = b

Multiply both sides by a and add a^2 - 2ab to both sides:

2a^2 - 2ab = a^2 - ab

Take 2 as a common factor:

2(a^2 - ab) = a^2 - ab

Divide by a^2 - ab:

2 = 1

Multiply by 2:

4 = 2

Add 3 to both sides:

7 = 5

3. A rectangle with sides of lengths x and y has perimeter P. Prove that the largest area of this rectangle is given when the rectangle is a square.

4. Two pipes fill a swimming pool in 11 1/9 hours (eleven hours and one ninth of an hour) together. One pipe can fill the pool in 5 hours less than the other pipe. Find out how much times it takes each pipe to fill up the swimming pool separately.

5. A pharmacist has two iodine solutions: one with a concentration of 30% and one with a concentration of 80%. How much of each solution the pharmacist needs to create 8 liters of a 50% solution?

6. Mary needs to refuel her car with 60 liters of gasoline and get back home. She has two choices: drive to a gas station 20 km away from her house which charges $1.16/liter or drive to a gas station adjacent to her house which charges $1.24/liter. Mary's car's mileage is 10 km/liter. Which is cheaper for Mary?

7. A brother and a sister are mowing the lawn on their backyard. It takes the brother 15 minutes longer to mow the lawn than it takes his sister. Together they mow the lawn in 56 minutes. How long does it take each of them to mow the lawn?

8. and say that is equal to

Exponent

What is an exponent?
What is a power?
Why is a power used?
Every topics must consider the 4 basic operations, what are these?
Evaluate and explain your approach for the following: (^ means power)

1. 5^0
2. 3^2 x 3^5
3. 8^0
4. b^1
5. k^0
6. m^5 x m^8
7. 4^7 x 2^6
8. 5^11 + 5^4

A world of no fractions

What is a fraction?
Is fraction necessary?
How will the world be without fractions?
Do you use fractions in your everyday routine?
What are the 4 basic operators?
Are these true or false? Explain.

1. 3/(7x) + 2/(7x) = 5 /(14x)
2. 3/(7x) + 2/(7x) = 5 /(7x)
3. 4/(9) = 4 x 1/9
4.

Equation

What is an equation?
What is a term?
What are connectors in an equation?
What is an expression?
Which can be solve and why an equation or an expression or both?
Can all equations be solved? Explain.
Give examples of life scenario and then someone formulate the equation or expression for it.

Curve

What is a curve?
What is a line?
How is a line different from a curve?
Give examples of something that has constant change in life?
Give examples of something that has varying changes in life?
Why and How are graphs connected to equations?
In a graph what is the most important things to observe?

Trigs

What is trigs about?
Why is trigs important to life?
Give examples of trigs used in life?
What is the difference between 3D trigs and 2D trigs?
Which trigs do you enjoy or used more 2D or 3D?
Give examples of how you use 3D trigs?